Multivector

A multivector is the result of a product defined for elements in a vector space V. A vector space with a linear product operation between elements of the space is called an algebra; examples are matrix algebra and vector algebra.^[1][2][3] The algebra of multivectors is constructed using the wedge product ∧ and is related to the exterior algebra of differential forms.^[4]

The set of multivectors on a vector space V is graded by the number of basis vectors that form a basis multivector. A multivector that is the product of p basis vectors is called a grade p multivector, or a p-vector. The linear combination of basis p-vectors forms a vector space denoted as Λ^p(V). The maximum grade of a multivector is the dimension of the vector space V.

The product of a p-vector and a k-vector is a (k + p)-vector so the set of linear combinations of all multivectors on V is an associative algebra, which is closed with respect to the wedge product. This algebra, denoted by Λ(V), is called the exterior algebra of V.^[5]

Wedge product

The wedge product operation used to construct multivectors is linear, associative and alternating, which reflect the properties of the determinant. This means for vectors u, v and w in a vector space V and for scalars α, β, the wedge product has the properties,

• Linear: $\mathbf{u}\wedge(\alpha\mathbf{v}+\beta\mathbf{w})=\alpha\mathbf{u}\wedge\mathbf{v}+\beta\mathbf{u}\wedge\mathbf{w};$
• Associative: $(\mathbf{u}\wedge\mathbf{v})\wedge\mathbf{w}=\mathbf{u}\wedge(\mathbf{v}\wedge\mathbf{w})=\mathbf{u}\wedge\mathbf{v}\wedge\mathbf{w};$
• Alternating: $\mathbf{u}\wedge\mathbf{v}=-\mathbf{v}\wedge\mathbf{u}, \quad\mathbf{u}\wedge\mathbf{u}=0.$

The product of p vectors is called a grade p multivector, or a p-vector. The maximum grade of a multivector is the dimension of the vector space V.

The linearity of the wedge product allows a multivector to be defined as the linear combination of basis multivectors. There are (n
p) basis p-vectors in an n-dimensional vector space.^[4]

Area and volume

The p-vector obtained from the wedge product of p separate vectors in an n-dimensional space has components that define the projected (p − 1)-volumes of the p-parallelotope spanned by the vectors. The square root of the sum of the squares of these components defines the volume of the p-parallelotope.^[4][6]

The following examples show that a bivector in two dimensions measures the area of a parallelogram, and the magnitude of a bivector in three dimensions also measures the area of a parallelogram. Similarly, a three-vector in three dimensions measures the volume of a parallelepiped.

It is easy to check that the magnitude of a three-vector in four dimensions measures the volume of the parallelepiped spanned by these vectors.

Multivectors in R²

Properties of multivectors can be seen by considering the two dimensional vector space V = R². Let the basis vectors be e₁ and e₂, so u and v are given by

$\mathbf{u}=u_1\mathbf{e}_1+u_2\mathbf{e}_2,\quad \mathbf{v}=v_1\mathbf{e}_1+v_2\mathbf{e}_2,$

and the multivector u ∧ v, also called a bivector, is computed to be

$\mathbf{u} \wedge \mathbf{v} =\begin{vmatrix} u_1 & v_1 \\ u_2 & v_2\end{vmatrix} \mathbf{e}_1\wedge\mathbf{e}_2.$

The vertical bars denote the determinant of the matrix, which is the area of the parallelogram spanned by the vectors u and v. The magnitude of u ∧ v is the area of this parallelogram. Notice that because V has dimension two the basis bivector e₁ ∧ e₂ is the only multivector in ΛV.

The relationship between the magnitude of a multivector and the area or volume spanned by the vectors is an important feature in all dimensions. Furthermore, the linear functional version of a multivector that computes this volume is known as a differential form.

Multivectors in R³

More features of multivectors can be seen by considering the three dimensional vector space V = R³. In this case, let the basis vectors be e₁, e₂, and e₃, so u, v and w are given by

$\mathbf{u}=u_1\mathbf{e}_1+u_2\mathbf{e}_2 +u_3\mathbf{e}_3 ,\quad \mathbf{v}=v_1\mathbf{e}_1+v_2\mathbf{e}_2+v_3\mathbf{e}_3, \quad \mathbf{w}=w_1\mathbf{e}_1+w_2\mathbf{e}_2+w_3\mathbf{e}_3,$

and the bivector u ∧ v is computed to be

$\mathbf{u} \wedge \mathbf{v} =\begin{vmatrix} u_2 & v_2 \\ u_3 & v_3\end{vmatrix} \mathbf{e}_2\wedge\mathbf{e}_3 + \begin{vmatrix} u_1 & v_1 \\ u_3 & v_3\end{vmatrix} \mathbf{e}_1\wedge\mathbf{e}_3 +\begin{vmatrix} u_1 & v_1 \\ u_2 & v_2\end{vmatrix} \mathbf{e}_1\wedge\mathbf{e}_2.$

The components of this bivector are the same as the components of the cross product. The magnitude of this bivector is the square root of the sum of the squares of its components.

This shows that the magnitude of the bivector u ∧ v is the area of the parallelogram spanned by the vectors u and v as it lies in the three-dimensional space V. The components of the bivector are the projected areas of the parallelogram on each of the three coordinate planes.

Notice that because V has dimension three, there is one basis three-vector in ΛV. Compute the three-vector

$\mathbf{u}\wedge\mathbf{v}\wedge\mathbf{w}=\begin{vmatrix} u_1 & v_1 &w_1\\ u_2 & v_2& w_2\\u_3&v_3&w_3\end{vmatrix} \mathbf{e}_1\wedge\mathbf{e}_2\wedge\mathbf{e}_3.$

This shows that the magnitude of the three-vector u ∧ v ∧ w is the volume of the parallelepiped spanned by the three vectors u, v and w.

In higher-dimensional spaces, the component three-vectors are projections of the volume of a parallelepiped onto the coordinate three-spaces, and the magnitude of the three-vector is the volume of the parallelepiped as it sits in the higher-dimensional space.

Grassmann coordinates

In this section, we consider multivectors on a projective space Pⁿ, which provide a convenient set of coordinates for lines, planes and hyperplanes that have properties similar to the homogeneous coordinates of points, called Grassmann coordinates.^[7]

Points in a real projective space Pⁿ are defined to be lines through the origin of the vector space Rⁿ⁺¹. For example, the projective plane P² is the set of lines through the origin of R³. Thus, multivectors defined on Rⁿ⁺¹ can be viewed as multivectors on Pⁿ.

A convenient way to view a multivector on Pⁿ is to examine it in an affine component of Pⁿ, which is the intersection of the lines through the origin of Rⁿ⁺¹ with a selected hyperplane, such as H: x_n+1 = 1. Lines through the origin of R³ intersect the plane E: z = 1 to define an affine version of the projective plane that only lacks the points z = 0, called the points at infinity.

Multivectors on P²

Points in the affine component E: z = 1 of the projective plane have coordinates x = (x, y, 1). A linear combination of two points p = (p₁, p₁, 1) and q = (q₁, q₁, 1) defines a plane in R³ that intersects E in the line joining p and q. The multivector p ∧ q defines a parallelogram in R³ given by

$\mathbf{p} \wedge \mathbf{q} =(p_2 - q_2)\mathbf{e}_2\wedge\mathbf{e}_3 + (p_1- q_1) \mathbf{e}_1\wedge\mathbf{e}_3 +(p_1 q_2- q_1 p_2)\mathbf{e}_1\wedge\mathbf{e}_2.$

Notice that substitution of αp + βq for p multiplies this multivector by a constant. Therefore, the components of p ∧ q are homogeneous coordinates for the plane through the origin of R³.

The set of points x = (x, y, 1) on the line through p and q is the intersection of the plane defined by p ∧ q with the plane E: z = 1. These points satisfy x ∧ p ∧ q = 0, that is,

$\mathbf{x}\wedge\mathbf{p} \wedge \mathbf{q} = (x\mathbf{e}_1+y\mathbf{e}_2+\mathbf{e}_3)\wedge \big( (p_2 - q_2)\mathbf{e}_2\wedge\mathbf{e}_3 + (p_1- q_1) \mathbf{e}_1\wedge\mathbf{e}_3 +(p_1 q_2- q_1 p_2)\mathbf{e}_1\wedge\mathbf{e}_2\big)=0,$

which simplifies to the equation of a line

$\lambda: x(p_2 - q_2) + y(p_1- q_1)+ (p_1 q_2- q_1 p_2)=0.$

This equation is satisfied by points x = αp + βq for real values of α and β.

The three components of p ∧ q that define the line λ are called the Grassmann coordinates of the line. Because three homogeneous coordinates define both a point and a line, the geometry of points is said to be dual to the geometry of lines in the projective plane. This is called the principle of duality.

Multivectors on P³

Three dimensional projective space, P³ consists of all lines through the origin of R⁴. Let the three dimensional hyperplane, H: w = 1, be the affine component of projective space defined by the points x = (x, y, z, 1). The multivector p ∧ q ∧ r defines a parallelepiped in R⁴ given by

$\mathbf{p}\wedge\mathbf{q}\wedge\mathbf{r}=\begin{vmatrix} p_2 & q_2 &r_2\\ p_3 & q_3& r_3\\1&1&1\end{vmatrix}\mathbf{e}_2\wedge\mathbf{e}_3\wedge\mathbf{e}_4 + \begin{vmatrix} p_1 & q_1 &r_1\\ p_3 & q_3& r_3\\1&1&1\end{vmatrix}\mathbf{e}_1\wedge\mathbf{e}_3\wedge\mathbf{e}_4 + \begin{vmatrix} p_1 & q_1 &r_1\\ p_2 & q_2& r_2\\1&1&1\end{vmatrix}\mathbf{e}_1\wedge\mathbf{e}_2\wedge\mathbf{e}_4 + \begin{vmatrix} p_1 & q_1 &r_1\\ p_2 & q_2& r_2\\ p_3 & q_3& r_3\end{vmatrix} \mathbf{e}_1\wedge\mathbf{e}_2\wedge\mathbf{e}_3.$

Notice that substitution of αp + βq + γr for p multiplies this multivector by a constant. Therefore, the components of p ∧ q ∧ r are homogeneous coordinates for the 3-space through the origin of R⁴.

A plane in the affine component H: w = 1 is the set of points x = (x, y, z, 1) in the intersection of H with the 3-space defined by p ∧ q ∧ r. These points satisfy x ∧ p ∧ q ∧ r = 0, that is,

$\mathbf{x}\wedge\mathbf{p} \wedge \mathbf{q}\wedge\mathbf{r} = (x\mathbf{e}_1+y\mathbf{e}_2+z\mathbf{e}_3 +\mathbf{e}_4)\wedge \mathbf{p}\wedge\mathbf{q}\wedge\mathbf{r} = 0 ,$

which simplifies to the equation of a plane

$\lambda: x\begin{vmatrix} p_2 & q_2 &r_2\\ p_3 & q_3& r_3\\1&1&1\end{vmatrix} + y \begin{vmatrix} p_1 & q_1 &r_1\\ p_3 & q_3& r_3\\1&1&1\end{vmatrix}+ z\begin{vmatrix} p_1 & q_1 &r_1\\ p_2 & q_2& r_2\\1&1&1\end{vmatrix}+ \begin{vmatrix} p_1 & q_1 &r_1\\ p_2 & q_2& r_2\\ p_3 & q_3& r_3\end{vmatrix} =0.$

This equation is satisfied by points x = αp + βq + γr for real values of α, β and γ.

The four components of p ∧ q ∧ r that define the plane λ are called the Grassmann coordinates of the plane. Because four homogeneous coordinates define both a point and a plane in projective space, the geometry of points is dual to the geometry of planes.

A line as the join of two points: In projective space the line λ through two points p and q can be viewed as the intersection of the affine space H: w = 1 with the plane x = αp + βq in R⁴. The multivector p ∧ q provides homogeneous coordinates for the line

$\lambda: \mathbf{p} \wedge \mathbf{q} = (p_1\mathbf{e}_1+p_2\mathbf{e}_2+p_3\mathbf{e}_3 +\mathbf{e}_4)\wedge (q_1\mathbf{e}_1+q_2\mathbf{e}_2+q_3\mathbf{e}_3 +\mathbf{e}_4),$

$\qquad =\begin{vmatrix} p_1 & q_1\\ 1 & 1 \end{vmatrix}\mathbf{e}_1\wedge\mathbf{e}_4 + \begin{vmatrix} p_2 & q_2\\ 1 & 1 \end{vmatrix}\mathbf{e}_2\wedge\mathbf{e}_4 + \begin{vmatrix} p_3 & q_3\\ 1 & 1 \end{vmatrix}\mathbf{e}_3\wedge\mathbf{e}_4+ \begin{vmatrix} p_2 & q_2\\ p_3 & q_3 \end{vmatrix}\mathbf{e}_2\wedge\mathbf{e}_3+\begin{vmatrix} p_3 & q_3\\ p_1 & q_1 \end{vmatrix}\mathbf{e}_3\wedge\mathbf{e}_1+\begin{vmatrix} p_1 & q_1\\ p_2 & q_2\end{vmatrix}\mathbf{e}_1\wedge\mathbf{e}_2.$

These are known as the Plücker coordinates of the line, though they are also an example of Grassmann coordinates.

A line as the intersection of two planes: A line μ in projective space can also be defined as the set of points x that form the intersection of two planes π and ρ defined by grade three multivectors, so the points x are the solutions to the linear equations

$\mu: \mathbf{x}\wedge \pi = 0, \mathbf{x}\wedge \rho = 0.$

In order to obtain the Plucker coordinates of the line μ, map the multivectors π and ρ to their dual point coordinates using the Hodge star operator,^[4]

$\mathbf{e}_1 = {\star}(\mathbf{e}_2\wedge\mathbf{e}_3 \wedge\mathbf{e}_4), -\mathbf{e}_2 = {\star}(\mathbf{e}_1\wedge\mathbf{e}_3 \wedge\mathbf{e}_4), \mathbf{e}_3 = {\star}(\mathbf{e}_1\wedge\mathbf{e}_2 \wedge\mathbf{e}_4), -\mathbf{e}_4 = {\star}(\mathbf{e}_1\wedge\mathbf{e}_2 \wedge\mathbf{e}_3),$

then

${\star}\pi = \pi_1\mathbf{e}_1 + \pi_2\mathbf{e}_2 + \pi_3\mathbf{e}_3 + \pi_4\mathbf{e}_4, \quad {\star}\rho = \rho_1\mathbf{e}_1 + \rho_2\mathbf{e}_2 + \rho_3\mathbf{e}_3 + \rho_4\mathbf{e}_4 .$

So, the Plücker coordinates of the line μ are given by

$\mu: ({\star}\pi)\wedge({\star}\rho) =\begin{vmatrix} \pi_1 & \rho_1\\ \pi_4 & \rho_4 \end{vmatrix}\mathbf{e}_1\wedge\mathbf{e}_4 + \begin{vmatrix} \pi_2 & \rho_2\\ \pi_4 & \rho_4 \end{vmatrix}\mathbf{e}_2\wedge\mathbf{e}_4 + \begin{vmatrix} \pi_3 & \rho_3\\ \pi_4 & \rho_4\end{vmatrix}\mathbf{e}_3\wedge\mathbf{e}_4+ \begin{vmatrix} \pi_2 & \rho_2\\ \pi_3 & \rho_3\end{vmatrix}\mathbf{e}_2\wedge\mathbf{e}_3+\begin{vmatrix} \pi_3 & \rho_3\\ \pi_1 & \rho_1\end{vmatrix}\mathbf{e}_3\wedge\mathbf{e}_1+\begin{vmatrix} \pi_1 & \rho_1\\ \pi_2 & \rho_2\end{vmatrix}\mathbf{e}_1\wedge\mathbf{e}_2.$

Because the six homogeneous coordinates of a line can be obtained from the join of two points or the intersection of two planes, the line is said to be self dual in projective space.

Clifford product

W. K. Clifford combined multivectors with the inner product defined on the vector space, in order to obtain a general construction for hypercomplex numbers that includes the usual complex numbers and Hamilton's quaternions.^[8][9]

The Clifford product between two vectors u and v is linear and associative like the wedge product, and has the additional property that the multivector uv is coupled to the inner product u · v by Clifford's relation,

$\mathbf{u}\mathbf{v} + \mathbf{v}\mathbf{u} = -2\mathbf{u}\cdot\mathbf{v}.$

Clifford's relation preserves the alternating property for the product of vectors that are perpendicular. This can be seen for the orthogonal unit vectors e_i, i = 1, ..., n in Rⁿ. Clifford's relation yields

$\mathbf{e}_i\mathbf{e}_j + \mathbf{e}_j\mathbf{e}_i = -2\mathbf{e}_i\cdot\mathbf{e}_j = 0,$

therefore the basis vectors are alternating,

$\mathbf{e}_i\mathbf{e}_j = - \mathbf{e}_j\mathbf{e}_i, \quad i\neq j = 1, \ldots, n.$

In contrast to the wedge product, the Clifford product of a vector with itself is no longer zero. To see this compute the product,

$\mathbf{e}_i\mathbf{e}_i + \mathbf{e}_i\mathbf{e}_i = -2 \mathbf{e}_i\cdot\mathbf{e}_i = -2,$

which yields

$\mathbf{e}_i\mathbf{e}_i = -1,\quad i=1,\ldots, n.$

The set of multivectors constructed using Clifford's product yields an associative algebra known as a Clifford algebra. Inner products with different properties can be used to construct different Clifford algebras.^[10][11]

Geometric algebra

Multivectors play a central role in the mathematical formulation of physics known as geometric algebra. The term geometric algebra was used by E. Artin for matrix methods in projective geometry.^[12] It was D. Hestenes who used geometric algebra to describe the application of Clifford algebras to classical mechanics,^[13] This formulation was expanded to geometric calculus by D. Hestenes and G. Sobczyk,^[14] who provided new terminology for a variety of features in this application of Clifford algebra to physics. C. Doran and A. Lasenby show that Hestene's geometric algebra provides a convenient formulation for modern physics.^[15]

In geometric algebra, a multivector is defined to be the sum of different-grade k-blades, such as the summation of a scalar, a vector, and a 2-vector.^[16] A sum of only k-grade components is called a k-vector,^[17] or a homogeneous multivector.^[18]

The highest grade element in a space is called a pseudoscalar.

If a given element is homogeneous of a grade k, then it is a k-vector, but not necessarily a k-blade. Such an element is a k-blade when it can be expressed as the wedge product of k vectors. A geometric algebra generated by a 4-dimensional Euclidean vector space illustrates the point with an example: The sum of any two blades with one taken from the XY-plane and the other taken from the ZW-plane will form a 2-vector that is not a 2-blade. In a geometric algebra generated by a Euclidean vector space of dimension 2 or 3, all sums of 2-blades may be written as a single 2-blade.

Examples

Orientation defined by an ordered set of vectors.

Reversed orientation corresponds to negating the exterior product.

Geometric interpretation of grade n elements in a real exterior algebra for n = 0 (signed point), 1 (directed line segment, or vector), 2 (oriented plane element), 3 (oriented volume). The exterior product of n vectors can be visualized as any n-dimensional shape (e.g. n-parallelotope, n-ellipsoid); with magnitude (hypervolume), and orientation defined by that on its (n − 1)-dimensional boundary and on which side the interior is.^[19][20]

• 0-vectors are scalars;
• 1-vectors are vectors;
• 2-vectors are bivectors;
• (n − 1)-vectors are pseudovectors;
• n-vectors are pseudoscalars.

In the presence of a volume form (such as given an inner product and an orientation), pseudovectors and pseudoscalars can be identified with vectors and scalars, which is routine in vector calculus, but without a volume form this cannot be done without a choice.

In the algebra of physical space (the geometric algebra of Euclidean 3-space, used as a model of (3+1)-spacetime), a sum of a scalar and a vector is called a paravector, and represents a point in spacetime (the vector the space, the scalar the time).

Bivectors

A bivector is therefore an element of the antisymmetric tensor product of a tangent space with itself.

In geometric algebra, also, a bivector is a grade 2 element (a 2-vector) resulting from the wedge product of two vectors, and so it is geometrically an oriented area, in the same way a vector is an oriented line segment. If a and b are two vectors, the bivector a ∧ b has

• a norm which is its area, given by

  $\left\| \mathbf a \wedge \mathbf b \right\| = \left\| \mathbf{a} \right\| \, \left\| \mathbf{b} \right\| \, \sin(\phi_{a,b})$

• a direction: the plane where that area lies on, i.e., the plane determined by a and b, as long as they are linearly independent;
• an orientation (out of two), determined by the order in which the originating vectors are multiplied.

Bivectors are connected to pseudovectors, and are used to represent rotations in geometric algebra.

As bivectors are elements of a vector space Λ²V (where V is a finite-dimensional vector space with dim V = n), it makes sense to define an inner product on this vector space as follows. First, write any element F ∈ Λ²V in terms of a basis (e_i ∧ e_j)_{1 ≤ i < j ≤ n} of Λ²V as

$F = F^{ab} \mathbf{e}_a \wedge \mathbf{e}_b \quad (1 \le a < b \le n) ,$

where the Einstein summation convention is being used.

Now define a map G : Λ²V × Λ²V → R by insisting that

$G(F, H) := G_{abcd}F^{ab}H^{cd} ,$

where $G_{abcd}$ are a set of numbers.

Applications

Bivectors play many important roles in physics, for example, in the classification of electromagnetic fields.

See also

• Blade (geometry)
• Paravector

References

1. F. E. Hohn, Elementary Matrix Algebra, Dover Publications, 2011
2. H. Kishan, Vector Algebra and Calculus, Atlantic Publ., 2007
3. L. Brand, Vector Analysis, Dover Publications, 2006
4. H. Flanders, Differential Forms with Applications to the Physical Sciences, Academic Press, New York, NY, 1963
5. Baylis (1994). Theoretical methods in the physical sciences: an introduction to problem solving using Maple V. Birkhäuser. p. 234, see footnote. ISBN 0-8176-3715-X.
6. G. E. Shilov, Linear Algebra, (trans. R. A. Silverman), Dover Publications, 1977.
7. W. V. D. Hodge and D. Pedoe, Methods of Algebraic Geometry, Vol. 1, Cambridge Univ. Press, 1947
8. W. K. Clifford, "Preliminary sketch of bi-quaternions," Proc. London Math. Soc. Vol. 4 (1873) pp. 381-395
9. W. K. Clifford, Mathematical Papers, (ed. R. Tucker), London: Macmillan, 1882.
10. J. M. McCarthy, An Introduction to Theoretical Kinematics, pp. 62–5, MIT Press 1990.
11. O. Bottema and B. Roth, Theoretical Kinematics, North Holland Publ. Co., 1979
12. E. Artin, Geometric algebra, Interscience Publ., 1957
13. D. Hestenes, New Foundations for Classical Mechanics, Kluwer Academic Publishers, 1986.
14. D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, Springer Verlag, 1987
15. C. Doran and A. Lasenby, Geometric Algebra for Physicists, Cambridge Univ. Press, 2007.
16. Marcos A. Rodrigues (2000). "§1.2 Geometric algebra: an outline". Invariants for pattern recognition and classification. World Scientific. p. 3 ff. ISBN 981-02-4278-6.
17. R Wareham, J Cameron & J Lasenby (2005). "Applications of conformal geometric algebra in computer vision and graphics". In Hongbo Li; Peter J. Olver; Gerald Sommer. Computer algebra and geometric algebra with applications. Springer. p. 330. ISBN 3-540-26296-2.
18. Eduardo Bayro-Corrochano (2004). "Clifford geometric algebra: A promising framework for computer vision, robotics and learning". In Alberto Sanfeliu, José Francisco Martínez Trinidad, Jesús Ariel Carrasco Ochoa. Progress in pattern recognition, image analysis and applications. Springer. p. 25. ISBN 3-540-23527-2.
19. R. Penrose (2007). The Road to Reality. Vintage books. ISBN 0-679-77631-1.
20. J.A. Wheeler, C. Misner, K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. p. 83. ISBN 0-7167-0344-0.